Miscellaneous Mathematical Tools and Facts

Description

A random assortment of elementary mathematical formulas and calculators.

Examples

# Find the greatest common divisor of 57, 93 and 117. 
gcd(c(57, 93, 117))

Convert Decimal to Binary

Description

Convert a given decimal constant in the interval (0, 1) to the corresponding binary representation.

Usage

DecToBin(x, m = 32, format = "character") 

Arguments

Details

Default format is as a character string. Alternatives are plain which prints results to the device, and vector which output a binary vector.

Value

a vector containing the binary representation

Examples

x <- c(.81, .57, .333) 
DecToBin(x) # character output
DecToBin(x, format="vector") #  binary vector output
DecToBin(x, format="plain") 

Sieve of Eratosthenes

Description

The sieve of Eratosthenes is an ancient method for listing all prime numbers up to a given value n.

Usage

EratosthenesSieve(n)

Arguments

Details

The algorithm scans through the vector from 2 through n, eliminating all multiples of 2, then eliminating all multiples of the next smallest integer (3), and so on, until only the prime numbers less than n remain.

Value

a numeric vector containing all primes less than n.

Examples

EratosthenesSieve(100)

Binary Expansion of Positive Integers

Description

Convert positive integers to their corresponding binary representation.

Usage

IntDecToBin(x, m = 31) 

Arguments

Value

a matrix containing the binary representations

Examples

x <- c(81, 57, 333) 
IntDecToBin(as.integer(x)) 

Law of Cosines

Description

Use of the ancient law of cosines to determine the angle between two sides of a triangle, given lengths of all three sides. This is a generalization of Pythagoras' Theorem.

Usage

LawofCosines(sides)

Arguments

Value

a numeric constant giving the angle in between the sides corresponding to the first two components in sides. Result is expressed in degrees.

Examples

LawofCosines(c(3, 4, 5))

Law of Sines

Description

Use of the ancient law of sines to determine the angle opposite one side of a triangle, given the length of the opposite side as well as the angle opposite another side whose length is also known. Alternatively, one can find the length of the side opposite a given angle.

Usage

LawofSines(sides, angles, findAngle)

Arguments

Details

If findAngle is TRUE, sides should be of length 2 and the function will compute angle opposite the side with length given by the second element of sides. Otherwise, angles should be of length 2, and the function will calculate the length of the side opposite the angle corresponding to the second element of angles.

Value

a numeric constant giving the angle opposite the given side, if findAngle is TRUE, or giving the length of the side opposite the given angle, if findAngle is FALSE.

Examples

LawofSines(c(3, 4), 50)  # find angle opposite the side of length 4.
LawofSines(3, c(70, 50), findAngle = FALSE) # find length of side opposite the 50 degree angle

Matrix Power

Description

Implementation of efficient algorithm to compute the pth power of a matrix.

Usage

Matpower(X, p) 

Arguments

Value

a matrix containing the pth power of X.

References

Golub and Van Loan (1983) Matrix Computations. Algorithm 11.2-1. p. 393.

Maximum Run Length

Description

Calculate the maximum run length of of 0's in a binary vector.

Usage

MaxRunLength(x) 

Arguments

Value

the maximum run length of 0's

Examples

x <- c(0L, 1L, 1L, 1L, 0L, 0L, 1L, 1L, 0L, 0L, 1L)
MaxRunLength(x) # should be 2
MaxRunLength(1L - x) # should be 3
# Test of Mersenne Twishter

RNGkind("Mers") # ensure that default generator is in use
N <- 10000
x <- runif(N)
y <- DecToBin(x, format = "vector", m = 40)
MaxHeadRunsM <- apply(y, 1, MaxRunLength) # 0 run lengths (Heads)
MaxTailRunsM <- apply(1-y, 1, MaxRunLength) # 1 run lengths (Tails)
# distributions of Max 0 run lengths and Max 1 run lengths should match
boxplot(MaxHeadRunsM, MaxTailRunsM, axes=FALSE, main="Maximum Run Length")
axis(side=1, at=c(1, 2), label=c("Heads", "Tails"))
axis(2)
box()

# Comparison with Wichmann-Hill Generator

RNGkind("Wich")
x <- runif(N)
y <- DecToBin(x, format = "vector", m = 40)
MaxHeadRunsW <- apply(y, 1, MaxRunLength)
MaxTailRunsW <- apply(1-y, 1, MaxRunLength)
boxplot(MaxHeadRunsW, MaxTailRunsW, axes=FALSE, main="Maximum Run Length")
axis(side=1, at=c(1, 2), label=c("Heads", "Tails"))
axis(2)
box()
RNGkind("Mers") # restore default generator

Pseudorandom Number Testing via Random Forest

Description

Given a sequence of pseudorandom numbers, this function constructs a random forest prediction model for successive values, based on previous values up to a given lag. The ability of the random forest model to predict future values is inversely related to the quality of the sequence as an approximation to locally random numbers.

Usage

RNGtest(u, m=5) 

Arguments

Value

Side effect is a two way layout of graphs showing effectiveness of prediction on a training and a testing subset of data. Good predictions indicate a poor quality sequence.

Author(s)

W. John Braun

Examples

    x <- runif(200)
    RNGtest(x, m = 4)

Greatest Common Divisor

Description

Greatest common divisor or factor for all elements of a positive-integer-valued vector.

Usage

gcd(x)

Arguments

Details

The gcd is calculated using the Euclidean algorithm applied to successive pairs of the elements of x.

Value

a numeric constant containing the greatest common divisor.

Examples

x <- c(81, 57, 333)
gcd(x)

Modulus Cracker

Description

Infers the modulus m for a congruential random number generator.

Usage

mCracker(U, par = 1e6, maxit = 100) 

Arguments

Details

Basic idea: Let x(1) denote the minimum order statistic in x1, x2, ..., xn. Then the set (x1/m)/(x(1)/m)*(1:par) must contain at least one integer, and m is in that set, if par has been set correctly.

Value

a list consisting of

Examples

# set.seed(33663)
x <- runif(1000000)
Y <- mCracker(x)$m
log(Y, 2)  # should be 32

Simplest Case of LASSO Regression

Description

Simple linear regression estimators for slope, intercept and noise standard deviation with absolute value penalty on slope.

Usage

microLASSO(x, y, lambda)

Arguments

Value

a list consisting of

Examples

x <- runif(30)
y <- x + rnorm(30)
microLASSO(x, y, lambda = 0.5)

Alias Method for Generating Discrete Random Variates

Description

Efficient method for generating discrete random variates from any distribution with a finite number (N) of states. The method is described in detail in Section 10.1 of the given reference.

Usage

rAlias(n, P) 

Arguments

Value

numeric vector of containing n simulated values from the given discrete distribution

References

S. Ross (1990) A Course in Simulation, MacMillan.

Examples

  x <- rAlias(n = 100, P = c(1:5)/15)
  table(x)/100

Multiply-With-Carry Random Number Generator

Description

Basic implementation of the multiply-with-carry generator.

Usage

rMWC(n, par) 

Arguments

Value

a vector of n uniform random numbers

References

Marsaglia, G. (2003) Random Number Generators. Journal of Modern Applied Statistical Methods. 2(1):2.

Examples

x <- rMWC(58, c(5, 3, 6, 10))
summary(x)

Linear Congruential Generator

Description

Basic implementation of the linear congruential random number generator.

Usage

rlincong(n, seed, par) 

Arguments

Value

a vector of n uniform random numbers

Examples

x <- rlincong(1000, 6976, c(171, 0, 30269))
summary(x)

Vectorized Congruential Random Number Generator

Description

Basic vectorized implementation of the linear congruential generator for simulating uniform random numbers on the interval (0, 1).

Usage

rngV(n, seed, par) 

Arguments

Value

a vector of n uniform random numbers

References

Anderson, S.L. (1990) Random number generators on vector supercomputers and other advanced architectures. SIAM Review, 32(2):221-251.

Examples

x <- rngV(1000, 6976, c(171, 0, 30269, 10))
summary(x)

Shuffling Algorithm

Description

Implementation of a simple shuffling algorithm that can be used to randomly permute a given set of simulated random numbers.

Usage

shuffle(n, k = 100, x = runif(n))

Arguments

Value

a numeric vector